How to make a cylinder cap concave?
Another way you can do this is with Proportional Editing (O):



Steps taken in the gif


Add a cylinder (&#8679; ShiftA> Mesh > Cylinder)
Enter Edit Mode (&#8633; Tab)
Select the top and bottom faces (&#9096; Ctrl&#8633; Tab> Face to set the selection mode, then &#8679; Shift RMB to select)
Inset (I)
Select just the top face and inset again (I)
Add loopcuts (&#9096; CtrlR, then  MW to adjust number of cuts)
Select the middle ngon
Turn on Proportional Editing (O) and set the falloff type in the header
Move the face down on the Z axis (G, Z. Use  MW to adjust the falloff)
Add a subsurf modifier (&#9096; Ctrl1, where 1 refers to the number of subdivisions in the modifier. Can be any digit from 1 to 5)
Add supporting loopcuts (&#9096; CtrlR) to control the subsurf.